| L(s) = 1 | + 4-s − 4·5-s + 10·11-s + 8·19-s − 4·20-s + 5·25-s + 12·29-s − 12·41-s + 10·44-s + 49-s − 40·55-s + 8·59-s + 28·61-s − 64-s − 44·71-s + 8·76-s + 14·79-s − 40·89-s − 32·95-s + 5·100-s − 16·101-s + 12·109-s + 12·116-s + 47·121-s + 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s + 3.01·11-s + 1.83·19-s − 0.894·20-s + 25-s + 2.22·29-s − 1.87·41-s + 1.50·44-s + 1/7·49-s − 5.39·55-s + 1.04·59-s + 3.58·61-s − 1/8·64-s − 5.22·71-s + 0.917·76-s + 1.57·79-s − 4.23·89-s − 3.28·95-s + 1/2·100-s − 1.59·101-s + 1.14·109-s + 1.11·116-s + 4.27·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.143822448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.143822448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 93 T^{2} + 6440 T^{4} + 93 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 45 T^{2} - 4864 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 25 T^{2} - 8784 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.62665374846310945892663921420, −6.58369884052354703192891088167, −6.16171714343392920718920012888, −5.78848325552963764578088492281, −5.76879666220346968563388284045, −5.73897208425583635283495520832, −5.18918067963247172361566130165, −5.06724946473013750660831592861, −4.63804252167467743494770036993, −4.57114565165896804884669686442, −4.34415595427147739766254723521, −4.21405914033259949922599472121, −3.78843652470207451389055593263, −3.64559155923011983062783381475, −3.52880300812136237403988601367, −3.40122655571693001858281996416, −3.00626829534234449460379772150, −2.69448406907982314547603899489, −2.50136984998490771124591428495, −2.18178739634684364271039918437, −1.42711921378283049882110207328, −1.40335819461639824337226173344, −1.34018254750830761152498470876, −0.822506518160658038286962832966, −0.35127009267775384351503492811,
0.35127009267775384351503492811, 0.822506518160658038286962832966, 1.34018254750830761152498470876, 1.40335819461639824337226173344, 1.42711921378283049882110207328, 2.18178739634684364271039918437, 2.50136984998490771124591428495, 2.69448406907982314547603899489, 3.00626829534234449460379772150, 3.40122655571693001858281996416, 3.52880300812136237403988601367, 3.64559155923011983062783381475, 3.78843652470207451389055593263, 4.21405914033259949922599472121, 4.34415595427147739766254723521, 4.57114565165896804884669686442, 4.63804252167467743494770036993, 5.06724946473013750660831592861, 5.18918067963247172361566130165, 5.73897208425583635283495520832, 5.76879666220346968563388284045, 5.78848325552963764578088492281, 6.16171714343392920718920012888, 6.58369884052354703192891088167, 6.62665374846310945892663921420
